Ευκλείδειο Επίπεδο
Ευκλείδειον Επίπεδον Euclidean Plane Είναι ένα επίπεδο. Ετυμολογία Η ονομασία "Ευκλείδειο" σχετίζεται ετυμολογικά με το όνομα "Ευκλέιδης". Εισαγωγή Απόσταση The Euclidean distance between two points of the plane with Cartesian coordinates (x_1,y_1) and (x_2,y_2) is : d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}. This is the Cartesian version of Pythagoras' theorem. In three-dimensional space, the distance between points (x_1,y_1,z_1) and (x_2,y_2,z_2) is : d = \sqrt{(x_2-x_1)^2 + (y_2-y_1)^2+ (z_2-z_1)^2} which can be obtained by two consecutive applications of Pythagoras' theorem. Ευκλείδειοι Μετασχηματισμοί Μεταφορά (Translation) translation a set of points of the plane, preserving the distances and directions between them, is equivalent to adding a fixed pair of numbers (X'',''Y) to the Cartesian coordinates of every point in the set. That is, if the original coordinates of a point are (x'',''y), after the translation they will be : (x',y') = (x + X, y + Y).\, Κλιμάκωση (Scaling) To make a figure larger (μεγέθυνση) or smaller (Σμίκρυνση) is equivalent to multiplying the Cartesian coordinates of every point by the same positive number m''. If (''x,y'') are the coordinates of a point on the original figure, the corresponding point on the scaled figure has coordinates : (x',y') = (m x, m y).\, If ''m is greater than 1, the figure becomes larger; if m'' is between 0 and 1, it becomes smaller. Περιστροφή (Rotation) To rotate a figure counterclockwise around the origin by some angle \theta is equivalent to replacing every point with coordinates (''x,y'') by the point with coordinates (''x',y'), where : x'=x \cos \theta - y \sin \theta\, : y'=x \sin \theta + y \cos \theta.\, Thus: (x',y') = ((x \cos \theta - y \sin \theta\,) , (x \sin \theta + y \cos \theta\,)). Κατοπτρισμός (Reflection) If (x'', ''y) are the Cartesian coordinates of a point, then (−''x'', y'') are the coordinates of its reflection across the second coordinate axis (the Y axis), as if that line were a mirror. Likewise, (''x, −''y'') are the coordinates of its reflection across the first coordinate axis (the X axis). Γενικός Μετασχηματισμός (General transformations) The Euclidean transformations of the plane are the translations, rotations, scalings, reflections, and arbitrary compositions thereof. The result (x', y') of applying a Euclidean transformation to a point (x,y) is given by the formula : (x',y') = (x,y) A + b\, where A'' is a 2×2 matrix and ''b is a pair of numbers, that depend on the transformation; that is, : x' = x A_{1 1} + y A_{2 1} + b_{1}\, : y' = x A_{1 2} + y A_{2 2} + b_{2}.\, The matrix A'' must have orthogonal rows with same Euclidean length, that is, : A_{1 1} A_{2 1} + A_{1 2} A_{2 2} = 0\, and : A_{1 1}^2 + A_{1 2}^2 = A_{2 1}^2 + A_{2 2}^2. This is equivalent to saying that ''A times its transpose must be a diagonal matrix. If these conditions do not hold, the formula describes a more general affine transformation of the plane. The formulas define a translation if and only if A'' is the identity matrix. The transformation is a rotation around some point if and only if ''A is a rotation matrix, meaning that : A_{1 1}^2 + A_{1 2}^2 = A_{2 1}^2 + A_{2 2}^2 = A_{1 1} A_{2 2} - A_{2 1} A_{1 2} = 1. Μερικοί Μετασχηματισμοί In two-dimensional space 'R'2 linear maps are described by 2 × 2 real matrices. These are some examples: * rotation by 90 degrees counterclockwise: *: \mathbf{A}=\begin{pmatrix}0 & -1\\ 1 & 0\end{pmatrix} * rotation by θ'' degrees counterclockwise: *: \mathbf{A}= \begin{pmatrix}\cos\theta & -\sin\theta\\ \sin\theta & \cos\theta \end{pmatrix} * reflection against the ''x axis: *: \mathbf{A}= \begin{pmatrix} 1 & 0\\ 0 & -1 \end{pmatrix} * reflection against the y'' axis: *: \mathbf{A}=\begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix} * scaling by 2 in all directions: *: \mathbf{A}= \begin{pmatrix} 2 & 0\\ 0 & 2 \end{pmatrix} * horizontal shear mapping: *: \mathbf{A}=\begin{pmatrix}1 & m\\ 0 & 1\end{pmatrix} * squeeze mapping: *: \mathbf{A}=\begin{pmatrix}k & 0\\ 0 & 1/k\end{pmatrix} * projection onto the ''y axis: *: \mathbf{A}=\begin{pmatrix}0 & 0\\ 0 & 1\end{pmatrix}. Υποσημειώσεις Εσωτερική Αρθρογραφία *Ευκλείδειος Χώρος * Επίπεδο *[[]] Βιβλιογραφία * * Ιστογραφία *Ομώνυμο άρθρο στην Βικιπαίδεια *Ομώνυμο άρθρο στην Livepedia *[ ] *[ ] category: Επίπεδα